(1)

\begin{align*} \mod\left(x,a\right)+\mod\left(y,a\right) & =\sgn\left(a\right)\left\{ \mod\left(x\sgn\left(a\right),\left|a\right|\right)+\mod\left(y\sgn\left(a\right),\left|a\right|\right)\right\} \\ & =\sgn\left(a\right)\begin{cases} \mod\left(\sgn\left(a\right)\left(x+y\right),\left|a\right|\right)-\left|a\right| & \mod\left(x\sgn\left(a\right),\left|a\right|\right)+\mod\left(y\sgn\left(a\right),\left|a\right|\right)<0\\ \mod\left(\sgn\left(a\right)\left(x+y\right),\left|a\right|\right) & 0\leq\mod\left(x\sgn\left(a\right),\left|a\right|\right)+\mod\left(y\sgn\left(a\right),\left|a\right|\right)<\left|a\right|\\ \mod\left(\sgn\left(a\right)\left(x+y\right),\left|a\right|\right)+\left|a\right| & \left|a\right|\leq\mod\left(x\sgn\left(a\right),\left|a\right|\right)+\mod\left(y\sgn\left(a\right),\left|a\right|\right) \end{cases}\\ & =\sgn\left(a\right)\mod\left(\sgn\left(a\right)\left(x+y\right),\left|a\right|\right)+\sgn\left(a\right)\left|a\right|\cdot\begin{cases} -1 & \mod\left(x\sgn\left(a\right),\left|a\right|\right)+\mod\left(y\sgn\left(a\right),\left|a\right|\right)<0\\ 0 & 0\leq\mod\left(x\sgn\left(a\right),\left|a\right|\right)+\mod\left(y\sgn\left(a\right),\left|a\right|\right)<\left|a\right|\\ +1 & \left|a\right|\leq\mod\left(x\sgn\left(a\right),\left|a\right|\right)+\mod\left(y\sgn\left(a\right),\left|a\right|\right) \end{cases}\\ & =\mod\left(x+y,a\right)+a\cdot\begin{cases} -1 & \sgn\left(a\right)\left(\mod\left(x,a\right)+\mod\left(y,a\right)\right)<0\\ 0 & 0\leq\sgn\left(a\right)\left(\mod\left(x,a\right)+\mod\left(y,a\right)\right)<\left|a\right|\\ +1 & \left|a\right|\leq\sgn\left(a\right)\left(\mod\left(x,a\right)+\mod\left(y,a\right)\right) \end{cases}\\ & =\mod\left(x+y,a\right)+a\mzp_{0,1}\left(0,\left|a\right|;\sgn\left(a\right)\left(\mod\left(x,a\right)+\mod\left(y,a\right)\right)\right) \end{align*}

(2)

\begin{align*} \mod\left(x,a,b\right)+\mod\left(y,a,b\right) & =\sgn\left(a\right)\left\{ \mod\left(x\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)+\mod\left(y\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)\right\} \\ & =\sgn\left(a\right)\begin{cases} \mod\left(\sgn\left(a\right)\left(x+y\right),\left|a\right|,b\sgn\left(a\right)\right)-\left|a\right| & \mod\left(x\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)+\mod\left(y\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)<b\sgn\left(a\right)\\ \mod\left(\sgn\left(a\right)\left(x+y\right),\left|a\right|,b\sgn\left(a\right)\right) & b\sgn\left(a\right)\leq\mod\left(x\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)+\mod\left(y\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)<b\sgn\left(a\right)+\left|a\right|\\ \mod\left(\sgn\left(a\right)\left(x+y\right),\left|a\right|,b\sgn\left(a\right)\right)+\left|a\right| & b\sgn\left(a\right)+\left|a\right|\leq\mod\left(x\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)+\mod\left(y\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right) \end{cases}\\ & =\sgn\left(a\right)\mod\left(\sgn\left(a\right)\left(x+y\right),\left|a\right|,b\sgn\left(a\right)\right)+\sgn\left(a\right)\left|a\right|\cdot\begin{cases} -1 & \mod\left(x\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)+\mod\left(y\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)<b\sgn\left(a\right)\\ 0 & b\sgn\left(a\right)\leq\mod\left(x\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)+\mod\left(y\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)<b\sgn\left(a\right)+\left|a\right|\\ +1 & b\sgn\left(a\right)+\left|a\right|\leq\mod\left(x\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right)+\mod\left(y\sgn\left(a\right),\left|a\right|,b\sgn\left(a\right)\right) \end{cases}\\ & =\mod\left(x+y,a,b\right)+a\cdot\begin{cases} -1 & \sgn\left(a\right)\left(\mod\left(x,a,b\right)+\mod\left(y,a,b\right)\right)<b\sgn\left(a\right)\\ 0 & b\sgn\left(a\right)\leq\sgn\left(a\right)\left(\mod\left(x,a,b\right)+\mod\left(y,a,b\right)\right)<b\sgn\left(a\right)+\left|a\right|\\ +1 & b\sgn\left(a\right)+\left|a\right|\leq\sgn\left(a\right)\left(\mod\left(x,a,b\right)+\mod\left(y,a,b\right)\right) \end{cases}\\ & =\mod\left(x+y,a,b\right)+a\mzp_{0,1}\left(b\sgn\left(a\right),b\sgn\left(a\right)+\left|a\right|;\sgn\left(a\right)\left(\mod\left(x,a,b\right)+\mod\left(y,a,b\right)\right)\right) \end{align*}

(3)

\begin{align*} \mod\left(x,a\right)+\mod\left(-x,a\right) & =x-a\left\lfloor \frac{x}{a}\right\rfloor -x-a\left\lfloor \frac{-x}{a}\right\rfloor \\ & =a\left(\left\lceil \frac{x}{a}\right\rceil -\left\lfloor \frac{x}{a}\right\rfloor \right)\\ & =a\left(1-\delta_{0,\mod\left(\frac{x}{a},1\right)}\right)\\ & =a\left|\sgn\mod\left(\frac{x}{a},1\right)\right|\\ & =a\left|\sgn\mod\left(x,a\right)\right| \end{align*}

(4)

\begin{align*} \mod\left(x,a\right)-\mod\left(y,a\right) & =\mod\left(x,a\right)-\left\{ a\left|\sgn\mod\left(y,a\right)\right|-\mod\left(-y,a\right)\right\} \\ & =\mod\left(x,a\right)+\mod\left(-y,a\right)-a\left|\sgn\mod\left(y,a\right)\right|\\ & =\mod\left(x-y,a\right)+a\mzp_{0,1}\left(0,\left|a\right|;\sgn\left(a\right)\left(\mod\left(x,a\right)+\mod\left(-y,a\right)\right)\right)-a\left|\sgn\mod\left(y,a\right)\right| \end{align*}

(5)

\begin{align*} \mod\left(x,a\right)-\mod\left(-x,a\right) & =\mod\left(2x,a\right)+a\mzp_{0,1}\left(0,\left|a\right|;\sgn\left(a\right)\left(\mod\left(2x,a\right)\right)\right)-a\left|\sgn\mod\left(-x,a\right)\right|\\ & =\mod\left(2x,a\right)+a\mzp_{0,1}\left(0,\left|a\right|;\sgn\left(a\right)\left(\mod\left(2x,a\right)\right)\right)-a\left|\sgn\mod\left(x,a\right)\right| \end{align*}